The g factor of the 2+1 state of 172Hf was measured using the perturbed angular correlation technique in a static external magnetic field. The result, g(2+1) = 0.25(5), is discussed in relation to the systematics of the previously reported g factors in the Hf isotopes and compared with the predictions of several models. An interesting outcome of the analysis presented in this paper is the agreement between the calculated g factors within the interacting boson approximation (IBA) and the results of a large-scale shell model calculation. This agreement supports the emphasis in the IBA on the valence space. The undershooting of the empirical g factors near midshell in both models suggests that they underestimate the role of the saturation of collectivity, which is explicitly incorporated into a phenomenological model that agrees better with the data

Ahn, T.

Bauer, C.

Beausang, C. W.

Berant, Z.

Brenner, D. S.

Bunce, M.

Casperson, R. J.

Casten, R. F.

Darakchieva, B.

Dudziak, K.

Gurdal, G.

Heinz, A.

Henning, G.

LeBlanc, J.

McCutchan, E. A.

Meyer, D. A.

Oster, E.

Qian, J.

Regan, P. H.

Ross, T. J.

Schmidt, A.

Smith, M. K.

Terry, J. R.

Werner, V.

Williams, E.

Winkler, R.

Wolf, A.

English

Berant, Z

Oster, E

Casperson, RJ

Wolf, A

Werner, V

Heinz, A

Casten, RF

Gurdal, G

McCutchan, EA

Brenner, DS

Terry, JR

Winkler, R

Williams, E

Qian, J

Schmidt, A

Smith, MK

Ahn, T

Beausang, CW

Regan, PH

Ross, T

Bunce, M

Darakchieva, B

Meyer, DA

LeBlanc, J

Dudziak, K

Bauer, C

Henning, G

Surrey Research Insight

g factor of the 2(1)(+) state of Hf-172

University of Richmond

University of RichmondUR Scholarship RepositoryPhysics Faculty Publications Physics11-10-2009g Factor of the 2+1 State of 172HfZ. BerantE. OsterR. J. CaspersonA. WolfV. WernerSee next page for additional authorsFollow this and additional works at: http://scholarship.richmond.edu/physics-faculty-publicationsPart of the Nuclear CommonsThis Article is brought to you for free and open access by the Physics at UR Scholarship Repository. It has been accepted for inclusion in PhysicsFaculty Publications by an authorized administrator of UR Scholarship Repository. For more information, please contactscholarshiprepository@richmond.edu.Recommended CitationBerant, Z., E. Oster, R. J. Casperson, A. Wolf, V. Werner, A. Heinz, R. F. Casten, G. Gurdal, E. A. Mccutchan, D. S. Brenner, J. R. Terry,R. Winkler, E. Williams, J. Qian, A. Schmidt, M. K. Smith, T. Ahn, C. W. Beausang, P. H. Regan, T. Ross, M. Bunce, B. Darakchieva, D.A. Meyer, J. Leblanc, K. Dudziak, C. Bauer, and G. Henning. "g Factor of the 21+ State of 172Hf." Physical Review C 80, no. 5(November 10, 2009): 057303: 1 - 057303: 4. doi:10.1103/PhysRevC.80.057303.AuthorsZ. Berant, E. Oster, R. J. Casperson, A. Wolf, V. Werner, A. Heinz, R. F. Casten, G. Gurdal, E. A. McCutchan,D. S. Brenner, J. R. Terry, R. Winkler, E. Williams, J. Qian, A. Schmidt, M. K. Smith, T. Ahn, C. W. Beausang,P. H. Regan, T. J. Ross, M. Bunce, B. Darakchieva, D. A. Meyer, J. LeBlanc, K. Dudziak, C. Bauer, and G.HenningThis article is available at UR Scholarship Repository: http://scholarship.richmond.edu/physics-faculty-publications/41PHYSICAL REVIEW C 80, 057303 (2009)g factor of the 2+1 state of 172HfZ. Berant,1,2 E. Oster,2 R. J. Casperson,1 A. Wolf,1,2 V. Werner,1 A. Heinz,1 R. F. Casten,1 G. Gurdal,3 E. A. McCutchan,4D. S. Brenner,1,5 J. R. Terry,1 R. Winkler,1 E. Williams,1 J. Qian,1 A. Schmidt,1 M. K. Smith,1 T. Ahn,1 C. W. Beausang,6P. H. Regan,7 T. Ross,6,7 M. Bunce,1,7 B. Darakchieva,8 D. A. Meyer,8 J. LeBlanc,8K. Dudziak,8 C. Bauer,9 and G. Henning1,101Wright Nuclear Structure Laboratory, Yale University, New Haven, Connecticut 06520, USA2Nuclear Research Center Negev, Beer-Sheva 84190, Israel3Rutgers University, New Brunswick, New Jersey 08903, USA4Physics Division, Argonne National Laboratory, Argonne, Illinois 60439, USA5Chemistry Department, Clark University, Worcester, Massachusetts 01610, USA6Physics Department, University of Richmond, Richmond, Virginia 23173, USA7Department of Physics, University of Surrey, Guildford, GU2 7XH, United Kingdom8Rhodes College, Department of Physics, Memphis, Tennessee 38112, USA9Insitut fu¨r Kernphysik, Technische Universita¨t Darmstadt, Darmstadt D-64289, Germany10ENS CACHAN, France(Received 17 August 2009; published 10 November 2009)The g factor of the 2+1 state of 172Hf was measured using the perturbed angular correlation technique in a staticexternal magnetic field. The result, g(2+1 ) = 0.25(5), is discussed in relation to the systematics of the previouslyreported g factors in the Hf isotopes and compared with the predictions of several models. An interesting outcomeof the analysis presented in this paper is the agreement between the calculated g factors within the interactingboson approximation (IBA) and the results of a large-scale shell model calculation. This agreement supports theemphasis in the IBA on the valence space. The undershooting of the empirical g factors near midshell in bothmodels suggests that they underestimate the role of the saturation of collectivity, which is explicitly incorporatedinto a phenomenological model that agrees better with the data.DOI: 10.1103/PhysRevC.80.057303 PACS number(s): 21.10.Ky, 23.20.En, 27.70.+qA series of measurements of g factors of 2+1 states in proton-rich nuclei in the A= 160 region has recently been undertakenat the Wright Nuclear Structure Laboratory (WNSL) at YaleUniversity. Experimental results have been obtained for Er,Yb, and Hf isotopes [1–3]. An interesting trend, previouslyestablished only for the Pt isotopes [4], has been observedalso for the Yb and Hf isotopic chains. The main featureis that g factors in these isotopic chains have a very weakdependence on neutron number in the range from N = 96 toN = 108, weaker than the predictions of the hydrodynamicalmodel [5]. These experimental observations have triggeredseveral theoretical studies [6,7] that have attempted to explainthe results—either using a phenomenological approach [6]or via large-scale shell model microscopic calculations [7].The continuation of this series of experiments is therefore ofparticular interest to better establish the experimental observa-tions and their theoretical implications for the understandingof nuclear structure in this region.In this Brief Report, we present the results of a measurementof the g factor of the Iπ = 2+1 state in 172Hf. The half-life ofthis state was recently remeasured at WNSL [8] and foundto be 1.278(40) ns. Since this is a relatively long half-life,we used the integral perturbed angular correlation method.The experimental technique and setup were described in somedetail in Ref. [1]. The parent of 172Hf, 172Ta, was producedby the reaction 165Ho(12C,5n)172Ta, with a 17 pnA, 84 MeV12C beam from the tandem accelerator at WNSL. The 172Tanuclei were deposited on an aluminized tape collector andperiodically transported to the center of a superconducting coilcapable of producing magnetic fields of up to 6 T. The half-lifeof 172Ta is ∼37 min, and the cycle of source accumulation-transport-counting was chosen to be 80 min. Eight high-puritygermanium detectors, with relative efficiencies of 20–25%,were placed around the superconducting coil, at about 11 cmfrom its center; these detectors were used to measure the decayof the 172Ta source to excited states in the daughter 172Hf.Two runs were carried out, with two different magnetic fieldsettings: 3.80 and 5.55 T. The magnetic fields were determinedusing the field vs current calibration function supplied by themanufacturer of the superconducting coil and checked usinga calibrated Hall probe. The running time was approximately160 h for each field direction, so the total running time forboth field directions and both values of the magnetic fieldwas about 640 h. About 5 × 108 γ -γ coincidence eventswere recorded and analyzed off-line. In the sorting procedure,all events originating from pairs of detectors with the sameangular separation were sorted in the same spectrum. Specialcare was taken to ensure that the convention [9] of the signsof angles were observed. In Fig. 1, we present an example ofa total projection γ -γ coincidence matrix.To determine the g factor, we used the double ratioR(θ, B) =[I (θ, B)I (θ,−B)/I (−θ, B)I (−θ,−B)]1/2, (1)where I (θ, B) is the coincidence intensity at angle θ andexternal field B. The use of this relation has the advantagethat it eliminates the need of normalization for total integratedcurrent on the target for field up and field down, and the0556-2813/2009/80(5)/057303(4) 057303-1 ©2009 The American Physical SocietyBRIEF REPORTS PHYSICAL REVIEW C 80, 057303 (2009)FIG. 1. Total projection of a γ -γ coincidence matrix obtained in the current work. All lines labeled by their energy in keV are observedfollowing the β decay of 172Ta to 172Hf.different relative efficiencies of detector pairs cancel out andneed not be accounted for. The coincidence intensity I (θ, B)for a given spin sequence can be calculated using the formalismgiven by Frauenfelder and Steffen [10]. Of all the relevantγ -γ cascades, the maximum perturbation effect is obtained atdetector angles of 145◦ and 35◦ for 0+ → 2+ → 0+ spin cas-cades. We therefore used the double ratio for the 776–95 keV,0+2 → 2+1 → 0+1 cascade to extract the g factor of the 2+1state of 172Hf from the data. The eight germanium detectorswere set in such a way that 12 pairs were at 145◦(35◦), eightpairs were at 110◦(70◦), four pairs were at 105◦(75◦), andthe remaining four pairs were at 180◦. This setup maximizesthe statistics at 145◦ and 35◦. In addition to the 776–95 keVtransition, the double ratio R(θ, B) was also calculated forfive other cascades: 214–95, 858–95, 980–95, 1086–95, and1241–95 keV. The values of the double ratio calculated fromthe data for all six cascades considered in this experimentare given in Table I. We also extracted the double ratios at180◦ from the data. At this angle, the double ratio should be1.00 for all cascades, so the experimental results can be usedto check for systematic errors. To determine the g factor, wecalculated the functions R(145◦, B) vs g for a 0+ → 2+ → 0+cascade for B = 3.80 and 5.55 T. These two functions arepresented in Fig. 2. Comparison of the functions in Fig. 2 withthe experimental results given in Table I for the 776–95 keVcascade for both values of the magnetic field yields two valuesof the g factor:g(2+1 )3.80 Texp = 0.23(6), (2)g(2+1 )5.55 Texp = 0.28(8). (3)The weighted mean of these two values gives the finalexperimental value of this work:g(2+1 )exp = 0.25(5). (4)The result for the 1241–95 keV cascade was initiallyintended to be used to improve the error bar of the experimentalg factor, since the level at 1336 keV was believed to be a 0+4state [11]. However, comparison of the result of the doubleFIG. 2. Calculated double ratio R(145◦, B) vs the g factor, andthe experimental double ratio for the 776–95 keV cascade, for thetwo values of the magnetic field used in this experiment. The solidand dotted horizontal lines indicate the experimental values and theirerrors.057303-2BRIEF REPORTS PHYSICAL REVIEW C 80, 057303 (2009)TABLE I. Values of the double ratio R(θ, B) obtained from the coincidence data for several cascades of 172Hf.Cascade (keV) Spin sequence Angle (deg) Rexp (θ, 3.80 T) Rexp (θ, 5.55 T) Rcalc (θ, 3.80 T)a Rcalc (θ, 5.55 T)a776–95 0+2 – 2+1 – 0+1 145 1.36(8) 1.61(12) – –1241–95 (0)+4 – 2+1 – 0+1 145 0.87(8) 0.98(7) – –214–95 4+1 – 2+1 – 0+1 145 1.031(6) 1.017(7) 1.02(1) 1.03(1)858–95 2+2 – 2+1 – 0+1 145 1.17(5) 1.22(6) 1.07(1)b 1.09(1)b980–95 2+3 – 2+1 – 0+1 145 1.10(5) 1.09(5) 1.07(1)b 1.09(1)b1086–95 3+1 – 2+1 – 0+1 145 0.97(3) 0.97(3) 0.93(1)b 0.90(1)b776–95 0+2 – 2+1 – 0+1 180 1.08(7) 1.07(7) 1.00 1.001241–95 (0)+4 – 2+1 – 0+1 180 1.14(16) – 1.00 1.00214–95 4+1 – 2+1 – 0+1 180 1.021(8) 1.019(10) 1.00 1.00857–95 2+2 – 2+1 – 0+1 180 1.08(7) 1.07(7) 1.00 1.00980–95 2+3 – 2+1 – 0+1 180 1.04(8) 0.95(9) 1.00 1.001086–95 3+1 – 2+1 – 0+1 180 1.00(5) 1.01(11) 1.00 1.00aValues of Rcalc and its error bars, where given, were obtained using the value g(2+1 )exp = 0.25(5) (see text).bThe double ratio was calculated assuming pure E2 character for the first transition of the cascade.ratio for this cascade, given in Table I, with that of the776–95 keV cascade, shows clearly that this assignment iswrong. The results for the remaining four cascades in Table Iare used as a consistency check. In the sixth and seventhcolumns of the table, we present the values of the double ratiosfor these cascades, calculated using the g factor in Eq. (4). Wenote that the double ratio at 180◦ is expected to be 1.00 forall spin sequences. The agreement between the calculated andexperimental values in Table I indicates that systematic errors,if present, are within the statistical errors of the experiment.We now discuss the experimental values obtained in thisexperiment in the framework of several models and theoreticalcalculations: the two versions (rotational and vibrational) ofthe hydrodynamical model [5], the proton-neutron versionof the interacting boson approximation (IBA-2) [12], thephenomenological model of Jing-ye et al. [6], and the recentlyreported large-scale shell model calculation of Bao-An Binet al. [7]. The experimental result of the present experiment,the recently reported result for 170Hf [3], and the results for176,178,180Hf from the compilation of Stone [13] are presentedin Fig. 3, together with the predictions of the models andcalculations mentioned above. For the proton-neutron versionof the interacting boson approximation, we used the relation[12]g(2+1 ) =gπNπ + gνNνNπ + Nν , (5)where Nπ and Nν are the number of valence proton and neutronbosons, and gπ , gν are the boson g factors. For the latterparameters, we used the values gπ = 0.63 and gν = 0.05 [14].From Fig. 3, we see that the present result follows the trendobserved previously [3], namely, that the g factors of theHf isotopes are almost constant in the range N = 98–108.Although the error bars are too large to allow an unambiguousdifferentiation between the models and calculations presentedin Fig. 3, the phenomenological model of Zhang et al. [6]seems to give the best overall description of the data. Thismodel uses the concept of effective boson numbers [14] andassumes a reduction of the effective numbers of protons andneutrons across midshell. The results presented in this BriefReport, together with the previously reported data, confirm thevalidity of the concept of effective boson numbers. In Fig. 3, wealso present the effective neutron boson numbers Nνeff togetherwith the normal neutron boson numbers Nν . We clearly seethe saturation effect of the effective boson number, which isresponsible for the experimentally observed constant values ofthe g factors vs N and also for the well-known saturation ofthe B(E2) value in the same region [6].Another interesting feature of Fig. 3, which is also found forother nearby isotopic chains, is the good agreement betweenΝνFIG. 3. Systematics of g(2+1 ) data for the Hf isotopes. The resultfor N = 98 is from our previous measurement [3], the value atN = 100 is from this experiment, and the other three experimentalpoints are from Stone’s tabulation in Ref. [13]. The data arecompared with the predictions of several models: the rotational andvibrational limits of the hydrodynamical model [5], the IBA-2 [12],the phenomenological model of Zhang et al. [6], and the results ofa large-scale shell model microscopic calculation [7]. The numberof neutron bosons Nν (black circles) and the number of neutroneffective bosons (open circles), calculated with the phenomenologicalmodel [6], are also shown.057303-3BRIEF REPORTS PHYSICAL REVIEW C 80, 057303 (2009)the large-scale shell model calculation [7] and the predictionsof the interacting boson approximation in the mass rangeA = 150–200. We note, however, that the IBA-2 predictionswere obtained using the analytical formula given above, whichis valid only for the three limiting symmetries of this model.To better establish the comparison with the large-scale shellmodel calculation, we performed several numerical IBA-2calculations of the g factors in Fig. 3, using the code NPBOS[15] and parameters that were chosen following publishedtheoretical works for nuclei in this region [15,16]. Theresults confirmed the trend as well as the absolute valuespredicted by the analytical formula [Eq. (5)]. The similaritybetween the IBA-2 results and those from the large-scaleshell model calculations suggests that the IBA-2 capturesessential physics embodied in its focus on the valence spaceand in the different contributions of protons and neutrons.The fact that both calculations underestimate the data nearmidshell, compared to the phenomenological model, suggeststhat they underestimate the saturation aspects of nuclearcollectivity in regions with large numbers of valence nucleons.Further measurements of g factors in this region are underway, and it is expected they will trigger more theoreticalinvestigations.The authors are indebted to the staff of the Wright NuclearStructure Laboratory for the skillful operation of the tandemaccelerator and especially to Mr. Walter R. Garnett, Jr., forextensive technical support. A.W. and P.H.R. acknowledge thehospitality of the WNSL during the experiment. This work wassupported by the US DOE under Grant Nos. DE-FG02-91ER-40609, DE-AC02-06CH11357, and DE-FG02-88ER-40417;the US NSF under Grant No. PHY-0245018; the Yale FlintFund; and the STFC(UK) and YUUP under Project No.DPT2006K-120470.[1] Z. Berant et al., Phys. Rev. C 69, 034320 (2004).[2] A. Wolf et al., Phys. Rev. C 72, 027301 (2005).[3] A. Wolf et al., Phys. Rev. C 76, 047308 (2007).[4] A. E. Stuchbery et al., Phys. Rev. Lett. 76, 2246(1996).[5] W. Greiner, Nucl. Phys. 80, 417 (1966).[6] Jing-ye Zhang et al., Phys. Rev. C 73, 037301 (2006).[7] Bao-An Bian, Yao-Min Di, Gui-Lu Long, Yang Sun, Jing-yeZhang, and Javid A. Sheikh, Phys. Rev. C 75, 014312(2007).[8] M. R. Bunce, M.Sc. thesis, University of Surrey, 2009.[9] O. Dørum and B. Selsmark, Nucl. Instrum. Methods 97, 243(1971).[10] H. Frauenfelder and R. M. Steffen, in Alpha-, Beta-,and Gamma-Ray Spectroscopy, edited by K. Siegbahn(North-Holland, Amsterdam, 1965), p. 1151.[11] Table of Isotopes, 8th ed., edited by R. Firestone and V. Shirley(Wiley, New York, 1996).[12] M. Sambataro, O. Scholten, A. E. L. Dieperink, and G. Piccito,Nucl. Phys. A423, 333 (1984).[13] N. J. Stone, Table of Nuclear Magnetic Dipole and ElectricQuadrupole Moments, NNDC, 2001 (unpublished), http://www.nndc.bnl.gov/publications/preprints/nuclear-moments.pdf.[14] A. Wolf and R. F. Casten, Phys. Rev. C 36, 851 (1987).[15] G. Puddu, O. Scholten, and T. Otsuka, Nucl. Phys. A348, 109(1980).[16] R. Bijker, A. E. L. Dieperink, O. Scholten, and R. Spanhoff,Nucl. Phys. A344, 207 (1980).057303-4

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